Combining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities
6. Monte Carlo studies Combining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities
6. Monte Carlo studies
6.1 Populations constructed using artificial data
We
constructed four artificial populations; a description of each one is presented
in Table 6.1. Notice that in Populations I, III and IV the
N = 150
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0JaaGymaiaaiwdacaaIWaaaaa@3C72@
values of the
m
i
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS
baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C34@
were obtained by
sampling from a Poisson distribution, whereas in Population II by sampling from
a zero truncated negative binomial distribution. In addition, in Populations I
and II, the link-probabilities
p
i
j
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaaaa@3DDD@
were generated
by the Rasch model (3.2). In Population III they were generated by that model
but the random effects
β
j
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
aaa@3D9B@
were obtained by
sampling from a scaled Student’s T distribution with six degrees of freedom and
unit-variance instead of by sampling from the standard normal distribution.
Finally, in Population IV, the
p
i
j
(
k
)
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaGqaaOGaa8xgGiaabohaaaa@3FA0@
were generated
by the following latent class model proposed by Pledger (2000) in the context
of capture-recapture studies:
p
i j
(
k
)
= exp [
μ
(
k
)
+
α
i
(
k
)
+
β
j
(
k
)
+
(
α β
)
i j
(
k
)
] /
{
1 + exp [
μ
(
k
)
+
α
i
(
k
)
+
β
j
(
k
)
+
(
α β
)
i j
(
k
)
] }
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai
aadchadaqhaaWcbaGaamyAaiaadQgaaeaadaqadaqaaiaadUgaaiaa
wIcacaGLPaaaaaGccqGH9aqpciGGLbGaaiiEaiaacchadaWadaqaai
abeY7aTnaaCaaaleqabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaa
aOGaey4kaSIaeqySde2aa0baaSqaaiaadMgaaeaadaqadaqaaiaadU
gaaiaawIcacaGLPaaaaaGccqGHRaWkcqaHYoGydaqhaaWcbaGaamOA
aaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaakiabgUcaRmaabm
aabaGaeqySdeMaeqOSdigacaGLOaGaayzkaaWaa0baaSqaaiaadMga
caWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaaGccaGLBb
GaayzxaaaabaWaaiWaaeaacaaIXaGaey4kaSIaciyzaiaacIhacaGG
WbWaamWaaeaacqaH8oqBdaahaaWcbeqaamaabmaabaGaam4AaaGaay
jkaiaawMcaaaaakiabgUcaRiabeg7aHnaaDaaaleaacaWGPbaabaWa
aeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGaey4kaSIaeqOSdi2aa0
baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGc
cqGHRaWkdaqadaqaaiabeg7aHjabek7aIbGaayjkaiaawMcaamaaDa
aaleaacaWGPbGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMca
aaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2haaaaacaGGSaaaaa@82D9@
i = 1 , … , n ; j = 1 , 2 ,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
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ypa0JaaGymaiaacYcacqWIMaYscaaISaGaamOBaiaacUdacaWGQbGa
eyypa0JaaGymaiaacYcacaaIYaGaaiilaaaa@441A@
and
k = 1 , 2.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbGaey
ypa0JaaGymaiaacYcacaaIYaGaaiOlaaaa@3D34@
In this model
the people in
U
k
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS
baaSqaaiaadUgaaeqaaaaa@3A5B@
is divided into
two latent classes
(
j = 1 , 2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
aadQgacqGH9aqpcaaIXaGaaiilaiaaikdaaiaawIcacaGLPaaaaaa@3E0A@
according to
their propensities to be linked to the sampled clusters. The probability that a
randomly person in
U
k
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS
baaSqaaiaadUgaaeqaaaaa@3A5B@
is in class
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbaaaa@3954@
is
p
j
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa
aa@3CEF@
and the
p
i
j
(
k
)
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaGqaaOGaa8xgGiaabohaaaa@3FA0@
are the same for
all the people in the class
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaai
Olaaaa@3A06@
The
simulation experiment was carried out by repeatedly selecting
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbaaaa@395C@
samples from
each population by using the sampling design described in Section 2 with
initial sample size
n = 15.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey
ypa0JaaGymaiaaiwdacaGGUaaaaa@3C8A@
Thus, each time
that the value
m
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS
baaSqaaiaadMgaaeqaaaaa@3A71@
was included in
an initial sample, the value
x
i
j
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG4bWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaaaa@3DE5@
was obtained by
sampling from the Bernoulli distribution with mean
p
i
j
(
k
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaOGaaiOlaaaa@3E99@
Because of the
values assigned to
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3958@
and to the
parameters that appear in the expression of
p
i
j
(
k
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaOGaaiilaaaa@3E97@
the resulting
sampling rates were
f
1
≈
0.5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS
baaSqaaiaaigdaaeqaaOGaeyisISRaaGimaiaai6cacaaI1aaaaa@3E23@
and
f
2
≈
0.4.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS
baaSqaaiaaikdaaeqaaOGaeyisISRaaGimaiaai6cacaaI0aGaaiOl
aaaa@3ED5@
It is worth
noting that the characteristics of the populations and samples considered in
this study were not motivated by the ones of an actual study since this
sampling design has not been applied yet. Thus, the populations and samples
were constructed only with the purpose of analyzing the performance of the
proposed point and interval estimators.
Table 6.1
Parameters of the simulated populations
Table summary
This table displays the results of Parameters of the simulated populations. The information is grouped by Population I (appearing as row headers), Population II, Population III and Population IV (appearing as column headers).
Population I
Population II
Population III
Population IV
N
=
150
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@
N
=
150
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@
N
=
150
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@
N
=
150
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@
M
i
∼
Poisson
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa
aeiuaiaab+gacaqGPbGaae4CaiaabohacaqGVbGaaeOBaaaa@4963@
M
i
∼
Zero trunc
. neg
. binomial
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa
aeOwaiaabwgacaqGYbGaae4BaiaabccacaqG0bGaaeOCaiaabwhaca
qGUbGaae4yaiaab6cacaqGGaGaaeOBaiaabwgacaqGNbGaaeOlaiaa
bccacaqGIbGaaeyAaiaab6gacaqGVbGaaeyBaiaabMgacaqGHbGaae
iBaaaa@58BB@
M
i
∼
Poisson
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa
aeiuaiaab+gacaqGPbGaae4CaiaabohacaqGVbGaaeOBaaaa@4963@
M
i
∼
Poisson
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa
aeiuaiaab+gacaqGPbGaae4CaiaabohacaqGVbGaaeOBaaaa@4963@
E
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4145@
E
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4145@
E
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4145@
E
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4145@
V
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4156@
V
(
M
i
)
=
24
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGOmaiaaisdaaaa@420E@
V
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4156@
V
(
M
i
)
=
8
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa
eyypa0JaaGioaaaa@4156@
τ
1
=
1,209
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccqGH9aqpcaqGXaGaaeilaiaabkdacaqG
WaGaaeyoaaaa@4277@
τ
1
=
1,208
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaaikdacaaI
WaGaaGioaaaa@4293@
τ
1
=
1,209
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaaikdacaaI
WaGaaGyoaaaa@4294@
τ
1
=
1,209
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaaikdacaaI
WaGaaGyoaaaa@4294@
τ
2
=
400
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@
τ
2
=
400
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@
τ
2
=
400
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@
τ
2
=
400
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@
τ
=
1,609
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq
GH9aqpcaqGXaGaaeilaiaabAdacaqGWaGaaeyoaaaa@418A@
τ
=
1,608
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq
GH9aqpcaaIXaGaaiilaiaaiAdacaaIWaGaaGioaaaa@41A6@
τ
=
1,609
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq
GH9aqpcaaIXaGaaiilaiaaiAdacaaIWaGaaGyoaaaa@41A7@
τ
=
1,609
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq
GH9aqpcaaIXaGaaiilaiaaiAdacaaIWaGaaGyoaaaa@41A7@
α
i
(
k
)
=
c
k
M
i
1
/
4
+
0.001
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda
qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kiabg2da9maalaaabaGaam4yamaaBaaaleaacaWGRbaabeaaaOqaai
aad2eadaqhaaWcbaGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisda
aaaaaOGaey4kaSIaaGimaiaai6cacaaIWaGaaGimaiaaigdaaaaaaa@4B9E@
α
i
(
k
)
=
c
k
M
i
1
/
4
+
0.001
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda
qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kiabg2da9maalaaabaGaam4yamaaBaaaleaacaWGRbaabeaaaOqaai
aad2eadaqhaaWcbaGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisda
aaaaaOGaey4kaSIaaGimaiaai6cacaaIWaGaaGimaiaaigdaaaaaaa@4B9E@
α
i
(
k
)
=
c
k
M
i
1
/
4
+
0.001
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda
qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kiabg2da9maalaaabaGaam4yamaaBaaaleaacaWGRbaabeaaaOqaai
aad2eadaqhaaWcbaGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisda
aaaaaOGaey4kaSIaaGimaiaai6cacaaIWaGaaGimaiaaigdaaaaaaa@4B9E@
α
i
(
k
)
=
−
12
M
i
1
/
2
+
0.001
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda
qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kiabg2da9maalaaabaGaeyOeI0IaaGymaiaaikdaaeaacaWGnbWaa0
baaSqaaiaadMgaaeaadaWcgaqaaiaaigdaaeaacaaIYaaaaaaakiab
gUcaRiaaicdacaaIUaGaaGimaiaaicdacaaIXaaaaaaa@4BF2@
c
1
=
−
5.45
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI
0aGaaGynaaaa@41F4@
c
1
=
−
5.45
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI
0aGaaGynaaaa@41F4@
c
1
=
−
5.45
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI
0aGaaGynaaaa@41F4@
μ
( 1 )
= − 1.1 ;
μ
( 2 )
= − 1.2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
ahaaWcbeqaaiaacIcacaaIXaGaaiykaaaakiabg2da9iabgkHiTiaa
igdacaaIUaGaaGymaiaacUdacaaMe8UaaGPaVlabeY7aTnaaCaaale
qabaGaaiikaiaaikdacaGGPaaaaOGaeyypa0JaeyOeI0IaaGymaiaa
i6cacaaIYaaaaa@4F51@
c
2
=
−
5.85
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI
4aGaaGynaaaa@41F9@
c
2
=
−
5.85
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI
4aGaaGynaaaa@41F9@
c
2
=
−
5.85
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI
4aGaaGynaaaa@41F9@
β
1
( k )
= 1.5 ;
β
2
( k )
= 0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaaGymaaqaaiaacIcacaWGRbGaaiykaaaakiabg2da9iaa
igdacaGGUaGaaGynaiaacUdacaaMe8UaaGPaVlabek7aInaaDaaale
aacaaIYaaabaGaaiikaiaadUgacaGGPaaaaOGaeyypa0JaaGimaaaa
@4DB6@
β
j
(
k
)
∼
N
(
0,1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
rqqr1ngBPrgifHhDYfgaiuaakiab=XJi6iaad6eadaqadaqaaiaaic
dacaaISaGaaGymaaGaayjkaiaawMcaaaaa@4AB4@
β
j
(
k
)
∼
N
(
0
,
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
rqqr1ngBPrgifHhDYfgaiuaakiab=XJi6iaad6eadaqadaqaaiaaic
dacaGGSaGaaGymaaGaayjkaiaawMcaaaaa@4AAE@
β
j
(
k
)
∼
2
/
3
T
6
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
rqqr1ngBPrgifHhDYfgaiuaakiab=XJi6maakaaabaWaaSGbaeaaca
aIYaaabaGaaG4maaaaaSqabaGccaaMc8UaamivamaaBaaaleaacaaI
2aaabeaaaaa@4B31@
(
α
β
)
i
j
(
k
)
∼
N
(
0
,
1.25
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
abeg7aHjabek7aIbGaayjkaiaawMcaamaaDaaaleaacaWGPbGaamOA
aaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaarqqr1ngBPrgifH
hDYfgaiuaakiab=XJi6iaad6eadaqadaqaaiaaicdacaGGSaGaaGym
aiaac6cacaaIYaGaaGynamaaCaaaleqabaGaaGOmaaaaaOGaayjkai
aawMcaaaaa@51E4@
p
1
(
k
)
=
0.3
=
1
−
p
2
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaaigdaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGc
cqGH9aqpcaaIWaGaaiOlaiaaiodacqGH9aqpcaaIXaGaeyOeI0Iaam
iCamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaaaa@49C8@
From
each sample the following estimators of
τ
1
,
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccaGGSaGaeqiXdq3aaSbaaSqaaiaaikda
aeqaaaaa@3E78@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
were considered:
the proposed UMLEs
τ
^
1
(
U
)
,
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaOGaaiilaiqbes8a0zaajaWaa0baaSqaaiaaikdaaeaadaqada
qaaiaadwfaaiaawIcacaGLPaaaaaaaaa@4360@
and
τ
^
(
U
)
;
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaOGa
ai4oaaaa@3D93@
the proposed
CMLEs
τ
^
1
(
C
)
,
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaOGaaiilaiqbes8a0zaajaWaa0baaSqaaiaaikdaaeaadaqada
qaaiaadoeaaiaawIcacaGLPaaaaaaaaa@433C@
and
τ
^
(
C
)
;
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGa
ai4oaaaa@3D81@
the MLEs
τ
˜
1
,
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaacamaaBaaa
leaacaaIYaaabeaaaaa@3E96@
and
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acaaaa@3A39@
proposed by
Félix-Medina and Thompson (2004) and derived under the assumption of
homogeneous link-probabilities, and the Bayesian-assisted estimators
τ
⌣
1
,
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaafamaaBaaa
leaacaaIYaaabeaaaaa@3EAE@
and
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afaaaa@3A45@
proposed by
Félix-Medina and Monjardin (2006), derived also under the homogeneity
assumption and using the following initial distributions for
τ
1
,
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccaGGSaGaeqiXdq3aaSbaaSqaaiaaikda
aeqaaaaa@3E78@
and
α
i
(
k
)
= ln [
p
i
(
k
)
/
(
1 −
p
i
(
k
)
)
] ,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda
qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kiabg2da9iGacYgacaGGUbWaamWaaeaadaWcgaqaaiaadchadaqhaa
WcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaaaOqa
amaabmaabaGaaGymaiabgkHiTiaadchadaqhaaWcbaGaamyAaaqaam
aabmaabaGaam4AaaGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaaaa
aiaawUfacaGLDbaacaGGSaaaaa@4F9B@
where
p
i
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa
aa@3CEE@
is given by (3.2),
but setting
β
j
(
k
)
= 0 :
ξ (
τ
1
|
λ
1
) ∝
(
N
λ
1
)
τ
1
/
τ
1
!
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kiabg2da9iaaicdacaGG6aWaaSGbaeaacqaH+oaEdaqadaqaamaaei
aabaGaeqiXdq3aaSbaaSqaaiaaigdaaeqaaaGccaGLiWoacqaH7oaB
daWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacqGHDisTdaqada
qaaiaad6eacqaH7oaBdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGL
PaaadaahaaWcbeqaaiabes8a0naaBaaameaacaaIXaaabeaaaaaake
aacqaHepaDdaWgaaWcbaGaaGymaaqabaGccaGGHaaaaaaa@5736@
and
ξ
(
λ
1
)
∝
λ
1
a
1
−
1
exp
(
−
b
1
λ
1
)
;
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda
qadaqaaiabeU7aSnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMca
aiabg2Hi1kabeU7aSnaaDaaaleaacaaIXaaabaGaamyyamaaBaaame
aacaaIXaaabeaaliabgkHiTiaaigdaaaGcciGGLbGaaiiEaiaaccha
daqadaqaaiabgkHiTiaadkgadaWgaaWcbaGaaGymaaqabaGccqaH7o
aBdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacaGG7aaaaa@508A@
ξ
(
τ
2
|
λ
2
)
∝
λ
2
τ
2
/
τ
2
!
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai
abe67a4naabmaabaGaeqiXdq3aaSbaaSqaaiaaikdaaeqaaOWaaqqa
aeaacqaH7oaBdaWgaaWcbaGaaGOmaaqabaaakiaawEa7aaGaayjkai
aawMcaaiabg2Hi1kabeU7aSnaaDaaaleaacaaIYaaabaGaeqiXdq3a
aSbaaWqaaiaaikdaaeqaaaaaaOqaaiabes8a0naaBaaaleaacaaIYa
aabeaakiaacgcaaaaaaa@4CE9@
and
ξ
(
λ
2
)
∝
λ
2
a
2
−
1
exp
(
−
b
2
λ
2
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda
qadaqaaiabeU7aSnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMca
aiabg2Hi1kabeU7aSnaaDaaaleaacaaIYaaabaGaamyyamaaBaaame
aacaaIYaaabeaaliabgkHiTiaaigdaaaGcciGGLbGaaiiEaiaaccha
daqadaqaaiabgkHiTiaadkgadaWgaaWcbaGaaGOmaaqabaGccqaH7o
aBdaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGSaaaaa@5080@
and
ξ
(
α
i
(
k
)
|
θ
k
)
∝
exp
[
−
(
α
i
(
k
)
−
θ
k
)
2
/
2
σ
k
2
]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda
qadaqaamaaeiaabaGaeqySde2aa0baaSqaaiaadMgaaeaadaqadaqa
aiaadUgaaiaawIcacaGLPaaaaaaakiaawIa7aiabeI7aXnaaBaaale
aacaWGRbaabeaaaOGaayjkaiaawMcaaiabg2Hi1kGacwgacaGG4bGa
aiiCamaadmaabaWaaSGbaeaacqGHsisldaqadaqaaiabeg7aHnaaDa
aaleaacaWGPbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGa
eyOeI0IaeqiUde3aaSbaaSqaaiaadUgaaeqaaaGccaGLOaGaayzkaa
WaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiabeo8aZnaaDaaaleaa
caWGRbaabaGaaGOmaaaaaaaakiaawUfacaGLDbaaaaa@5C94@
and
ξ
(
θ
k
)
∝
exp
[
−
(
θ
k
−
μ
k
)
2
/
2
γ
k
2
]
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda
qadaqaaiabeI7aXnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMca
aiabg2Hi1kGacwgacaGG4bGaaiiCamaadmaabaWaaSGbaeaacqGHsi
sldaqadaqaaiabeI7aXnaaBaaaleaacaWGRbaabeaakiabgkHiTiab
eY7aTnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaamaaCaaale
qabaGaaGOmaaaaaOqaaiaaikdacqaHZoWzdaqhaaWcbaGaam4Aaaqa
aiaaikdaaaaaaaGccaGLBbGaayzxaaGaaiilaaaa@53F4@
where
a
1
= 1.0 ,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaaGymaiaac6cacaaIWaGaaiil
aaaa@3E19@
b
1
= 0.1 ,
a
2
= 6.0 ,
b
2
= 0.01 ,
μ
k
= − 3.5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGIbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaiaai6cacaaIXaGaaiil
aiaadggadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI2aGaaGOlai
aaicdacaGGSaGaamOyamaaBaaaleaacaaIYaaabeaakiabg2da9iaa
icdacaaIUaGaaGimaiaaigdacaGGSaGaeqiVd02aaSbaaSqaaiaadU
gaaeqaaOGaeyypa0JaeyOeI0IaaG4maiaai6cacaaI1aaaaa@5159@
and
σ
k
2
=
γ
k
2
= 9.0.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda
qhaaWcbaGaam4AaaqaaiaaikdaaaGccqGH9aqpcqaHZoWzdaqhaaWc
baGaam4AaaqaaiaaikdaaaGccqGH9aqpcaaI5aGaaiOlaiaaicdaca
GGUaaaaa@4482@
These values
assigned to the parameters of the initial distributions made them practically
non-informative. The Gaussian quadrature approximations (3.3) and (3.4) to the
probabilities
π
x
(
k
)
(
α
k
,
σ
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda
qhaaWcbaGaaCiEaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
kmaabmaabaGaaCySdmaaBaaaleaacaWGRbaabeaakiaaiYcacqaHdp
WCdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPaaaaaa@455E@
and
π
x
(
A
i
)
(
α
1
−
i
,
σ
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda
qhaaWcbaGaaCiEaaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaa
beaaaSGaayjkaiaawMcaaaaakmaabmaabaGaaCySdmaaDaaaleaaca
aIXaaabaGaeyOeI0IaamyAaaaakiaaiYcacqaHdpWCdaWgaaWcbaGa
aGymaaqabaaakiaawIcacaGLPaaaaaa@47CC@
were computed
using
q = 40
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbGaey
ypa0JaaGinaiaaicdaaaa@3BD9@
terms.
The
performance of an estimator
τ
^
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcaaaa@3A3A@
of
τ
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDca
GGSaaaaa@3ADA@
say, was
evaluated by means of its relative bias
(
r
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikaiaabk
hacqGHsislaaa@3AE3@
bias), the square root of its relative mean square error
(
r
−
mse
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada
GcaaqaaiaabkhacqGHsislcaqGTbGaae4CaiaabwgaaSqabaaakiaa
wIcacaGLPaaacaGGSaaaaa@3F63@
and the medians
of its relative estimation errors (mdre) and its absolute relative estimation
errors (mdare) defined by
r − bias
=
∑
1
r
(
τ
^
i
− τ
) /
(
r τ
) ,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaabkgacaqGPbGaaeyyaiaabohadaWcgaqaaiabg2da9maaqada
beWcbaGaaGymaaqaaiaadkhaa0GaeyyeIuoakmaabmaabaGafqiXdq
NbaKaadaWgaaWcbaGaamyAaaqabaGccqGHsislcqaHepaDaiaawIca
caGLPaaaaeaadaqadaqaaiaadkhacqaHepaDaiaawIcacaGLPaaaca
GGSaaaaaaa@4EE6@
r − mse
=
∑
1
r
(
τ
^
i
− τ
)
2
/
(
r
τ
2
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGcaaqaai
aabkhacqGHsislcaqGTbGaae4CaiaabwgaaSqabaGccqGH9aqpdaGc
aaqaamaalyaabaWaaabmaeqaleaacaaIXaaabaGaamOCaaqdcqGHri
s5aOWaaeWaaeaacuaHepaDgaqcamaaBaaaleaacaWGPbaabeaakiab
gkHiTiabes8a0bGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaO
qaamaabmaabaGaamOCaiabes8a0naaCaaaleqabaGaaGOmaaaaaOGa
ayjkaiaawMcaaaaaaSqabaGccaGGSaaaaa@5049@
mdre = median {
(
τ
^
i
− τ
) / τ
}
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaab2gacaqGKbGaaeOCaiaabwgacaqGGaGaaeypaiaabccacaqG
TbGaaeyzaiaabsgacaqGPbGaaeyyaiaab6gapaWaaiWaaeaadaWcga
qaamaabmaabaGafqiXdqNbaKaadaWgaaWcbaGaamyAaaqabaGccqGH
sislcqaHepaDaiaawIcacaGLPaaaaeaacqaHepaDaaaacaGL7bGaay
zFaaaaaa@4F0E@
and
mdrae = median {
|
(
τ
^
i
− τ
) / τ
| } ,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaab2gacaqGKbGaaeOCaiaabggacaqGLbGaaeiiaiaab2dacaqG
GaGaaeyBaiaabwgacaqGKbGaaeyAaiaabggacaqGUbWdamaacmaaba
WaaqWaaeaadaWcgaqaamaabmaabaGafqiXdqNbaKaadaWgaaWcbaGa
amyAaaqabaGccqGHsislcqaHepaDaiaawIcacaGLPaaaaeaacqaHep
aDaaaacaGLhWUaayjcSdaacaGL7bGaayzFaaGaaiilaaaa@53C4@
where
τ
^
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaBaaaleaacaWGPbaabeaaaaa@3B54@
was the value of
τ
^
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcaaaa@3A3A@
obtained in the
i
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW
baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@
sample, which in
the case of the point estimators was 10,000.
We also considered the following 95% CIs for the
τ
’
s
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDie
aacaWFzaIaae4CaiaacQdaaaa@3CA1@
the proposed
PLCIs and adjusted for extra-Poisson variation PLCIs; the proposed bootstrap
CIs based on
B = 100
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGcbGaey
ypa0JaaGymaiaaicdacaaIWaaaaa@3C61@
bootstrap
samples and constructed assuming a lognormal distribution for
τ
^
−
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcaiabgkHiTiabe27aUbaa@3CDF@
and estimating
V
^
(
τ
^
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGcaaqaai
qadAfagaqcamaabmaabaGafqiXdqNbaKaaaiaawIcacaGLPaaaaSqa
baaaaa@3CC9@
by Huber’s
proposal 2 estimator of scale with tuning value
d = 1.5 ;
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGKbGaey
ypa0JaaGymaiaac6cacaaI1aGaai4oaaaa@3D3F@
the design-based
Wald CIs obtained from the MLEs
τ
˜
1
,
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaacamaaBaaa
leaacaaIYaaabeaaaaa@3E96@
and
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acaaaa@3A39@
and proposed by
Félix-Medina and Thompson (2004), and the design-based Wald CIs obtained from
the Bayesian estimators
τ
⌣
1
,
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaafamaaBaaa
leaacaaIYaaabeaaaaa@3EAE@
and
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afaaaa@3A45@
and proposed by
Félix-Medina and Monjardin (2006). It is worth noting that the PLCIs and
adjusted PLCI were computed using the Venzon and Moolgavkar’s (1988) method or
an algorithm based on the definition of a PLCI when the first method failed to
find the endpoints of the intervals.
The performance of a CI was evaluated by its coverage
probability (cp), the mean of its relative lengths (mrl) and the median of its
relative lengths (mdrl) defined as the proportion of samples in which the
parameter was contained in the interval and the the mean and the median of the
lengths of the intervals divided by the value of the parameter, respectively.
Since carrying out a simulation study on the CIs using a large number of
replicated samples is a very time consuming task, we evaluated the performance
of the PLCIs for
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
and
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
using
r = 1,000
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey
ypa0JaaGymaiaacYcacaaIWaGaaGimaiaaicdaaaa@3DFB@
samples; that of
the PLCIs for
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
using
r = 500
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey
ypa0JaaGynaiaaicdacaaIWaaaaa@3C95@
samples and that
of the bootstrap CIs using
r = 250
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey
ypa0JaaGOmaiaaiwdacaaIWaaaaa@3C97@
samples. The
performance of the CIs based on the estimators derived under the homogeneity
assumption was evaluated by using
r = 10,000
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey
ypa0JaaGymaiaaicdacaGGSaGaaGimaiaaicdacaaIWaaaaa@3EB5@
samples. The
numerical study was carried out using the R software programming language [R
Development Core Team (2013)].
The
results of the study on the estimators of the population sizes are shown in
Table 6.2 and in Figures 6.1 and 6.2. The main outcomes are the following. The
distributions of the estimators UMLE
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D85@
and CMLE
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaaaa@3D73@
were more or
less symmetrical about
τ
1
;
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccaGG7aaaaa@3BDA@
thus, the two
measures of bias
(
r
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikaiaabk
hacqGHsislaaa@3AE3@
bias and mdre) showed similar values, as well as the two
measures of variability
(
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikamaaka
aabaGaaeOCaiabgkHiTiaab2gacaqGZbGaaeyzaaWcbeaaaaa@3DCC@
and mdare). Both
of these estimators performed acceptably well, except in Population III where
the estimator
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D85@
presented
moderate problems of bias and
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaaaa@3D73@
showed something
more serious problems of bias. The distributions of the estimators
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D86@
and
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaaaa@3D74@
were skewed to
the right with very long tails. This caused that the values of their
r
−
bias
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3DE2@
and
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaab2gacaqGZbGaaeyzaaaa@3D05@
tended to be
large. However, in terms of the medians of their relative errors (mdre), these
estimators presented moderate problems of bias in Populations I and III and
serious problems in Population IV. In terms of the medians of their absolute
relative errors (mdare) these estimators showed moderate problems of
instability in the first three populations and serious problems in the fourth
population. The distributions of the estimators
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaaaa
@3CCA@
and
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaaaa
@3CB8@
were similar to
those of the estimators
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D86@
and
τ
^
2
(
C
)
;
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaOGaai4oaaaa@3E3D@
thus, the
quantities
r
−
bias
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3DE2@
and
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca
qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@
were more
sensitive to large values than the quantities mdre and mdare. Both of these
estimators performed acceptably well in Populations I, II and IV; although in
this last population the values of their
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca
qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@
were large
because of the reasons previously indicated. In Population III both estimators
presented problems of bias.
Although
the deviation from the assumed Poisson distribution of the
M
i
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C14@
increased the
variability of all the proposed estimators, the increments were not large so
that we consider that they have some robust properties against deviations from
this assumption. The proposed estimators of
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
were in addition
robust to the deviation from the assumed Rasch model for the
p
i
j
(
1
)
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaaIXaaacaGLOaGaayzk
aaaaaGqaaOGaa8xgGiaabohaaaa@3F6B@
(although the
values of the
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca
qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@
of the
estimators of
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
were large,
those of the median of their absolute relative errors were not). The deviation
from the assumed normal distribution of the effects
β
j
(
k
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda
qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa
aaa@3D9B@
caused that all
the proposed estimators presented problems of overestimation. Neither of the
two types of proposed estimators UMLEs and CMLEs performed uniformly better
than the other, but the UMLEs performed in a greater number of cases slightly
better than the CMLEs.
Table 6.2
Relative biases, square roots of relative mean square errors and medians of relative errors and absolute relative errors of the
estimators of the population sizes
Table summary
This table displays the results of Relative biases. The information is grouped by Population
Sampling rates
Estimator (appearing as row headers), I, II, III, IV, XXXX
0.51, XXXX
0.4 and XXXX
0.5, calculated using XXXX and XXXX
units of measure (appearing as column headers).
Population
I
II
III
IV
Sampling rates
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.51
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.50
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.51
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.51
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
Estimator
r
b
i
a
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x
fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk
haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@
r
m
s
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa
qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa
aaWcbeaaaaa@3BF1@
m
d
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x
fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@
m
d
a
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x
fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@
r
b
i
a
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk
haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@
r
m
s
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa
qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa
aaWcbeaaaaa@3BF1@
m
d
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@
m
d
a
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@
r
b
i
a
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk
haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@
r
m
s
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa
aaWcbeaaaaa@3BF1@
m
d
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@
m
d
a
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@
r
b
i
a
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x
fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk
haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@
r
m
s
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa
qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa
aaWcbeaaaaa@3BF1@
m
d
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x
fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@
m
d
a
r
e
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x
fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2
gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@
Uncond. heter. MLEs
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
-0.01
0.06
-0.01
0.04
-0.00
0.08
-0.00
0.05
0.10
0.11
0.10
0.10
-0
.04
6
0.20
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc
dacaGGUaGaaGimaiaaisdadaqhaaWcbaGaaGOnaaqaaiaaicdacaGG
UaGaaGOmaiaaicdaaaaaaa@3F5E@
0.08
-0.03
0.05
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D23@
-0.06
0.26
-0.11
0.17
0.060.02
0.35
-0.01
0.16
0.16
0.43
0.07
0.18
0.04
2
15
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6
cacaaIWaGaaGinamaaDaaaleaacaaIYaaabaGaaGymaiaaiwdaaaaa
aa@3D49@
2.2
-0.19
0.25
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@
-0.02
0.08
-0.03
0.05
0.010.02
0.10
0.01
0.06
0.11
0.15
0.10
0.10
-0
.02
7
15
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc
dacaGGUaGaaGimaiaaikdadaqhaaWcbaGaaG4naaqaaiaaigdacaaI
1aaaaaaa@3DF5@
0.55
-0.06
0.08
Cond. heter. MLEs
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
-0.00
0.07
-0.01
0.05
0.01
0.07
0.00
0.05
0.18
0.19
0.17
0.17
-0.051.6
0.09
-0.05
0.07
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D11@
-0.04
0.26
-0.09
0.17
0.09
0.38
0.01
0.16
0.18
0.46
0.10
0.18
0.12
2
21
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6
cacaaIXaGaaGOmamaaDaaaleaacaaIYaaabaGaaGOmaiaaigdaaaaa
aa@3D45@
2.4
-0.14
0.23
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
-0.01
0.08
-0.02
0.05
0.03
0.11
0.01
0.06
0.18
0.22
0.16
0.16
-0
.00
2
23
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc
dacaGGUaGaaGimaiaaicdadaqhaaWcbaGaaGOmaaqaaiaaikdacaaI
Zaaaaaaa@3DED@
0.61
-0.06
0.08
Homogeneous MLEs
τ
˜
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@
-0.28
0.28
-0.28
0.28
-0.31
0.31
-0.31
0.31
-0.30
0.30
-0.30
0.30
-0.18
0.19
-0.18
0.18
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@
-0.40
0.40
-0.40
0.40
-0.40
0.40
-0.40
0.40
-0.40
0.40
-0.40
0.40
-0.30
0.32
-0.32
0.32
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aaaaa@39D6@
-0.31
0.31
-0.31
0.31
-0.33
0.33
-0.33
0.33
-0.32
0.33
-0.32
0.32
-0.21
0.22
-0.21
0.21
Homogeneous BEs
τ
⌣
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@
-0.28
0.28
-0.28
0.28
-0.31
0.31
-0.31
0.31
-0.30
0.30
-0.30
0.30
-0.18
0.19
-0.18
0.18
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@
-0.39
0.39
-0.39
0.39
-0.39
0.40
-0.39
0.39
-0.39
0.40
-0.39
0.39
-0.27
0.30
-0.29
0.29
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
baaaa@39E2@
-0.31
0.31
-0.31
0.31
-0.33
0.33
-0.33
0.33
-0.32
0.32
-0.32
0.32
-0.20
0.21
-0.20
0.20
Description for Figure 6.1
Figure showing six boxplot graphs for the values of the estimators of
τ
1
,
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaigdaaeqaaOGaaiilaiabes8a0naaBaaaleaacaaIYaaa
beaaaaa@3E0A@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BC@
in Populations I and II. See Table 6.1 for the parameter values of the simulated populations. Each graph is made of four boxplots, one for each of UMLE, CMLE, MLE homogeneous and Bayes homogeneous. For both populations, the distributions of the estimators UMLE and CMLE were more or less symmetrical about
τ
1
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaigdaaeqaaOGaaiOlaaaa@3B5F@
The distributions of the estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. For both populations, the medians of the distributions of the estimators UMLE and CMLE are close to
τ
2
.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaikdaaeqaaOGaaiOlaaaa@3B5F@
But, these distributions are skewed to the right with very long tails. The distributions of estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. Similar conclusions can be drawn for
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BB@
boxplots.
Description for Figure 6.2
Figure showing six boxplot graphs for the values of the estimators of
τ
1
,
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaigdaaeqaaOGaaiilaiabes8a0naaBaaaleaacaaIYaaa
beaaaaa@3E0A@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BC@
in Populations III and IV. See Table 6.1 for the parameter values of the simulated populations. Each graph is made of four boxplots, one for each of UMLE, CMLE, MLE homogeneous and Bayes homogeneous. For Population III, the distributions of the estimators UMLE and CMLE show a bias for
τ
1
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaigdaaeqaaOGaaiilaaaa@3B5D@
worse for CMLE. The distributions of the estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. For both populations, for
τ
2
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaikdaaeqaaOGaaiilaaaa@3B5D@
the distributions are skewed to the right with very long tails. This problem is worse for Population IV. The distributions of the estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. Similar conclusions can be drawn for
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BB@
boxplots.
With regard to the estimators derived under the
assumption of homogeneous
p
i
j
(
k
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0
baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk
aaaaaOGaaiilaaaa@3E97@
both the MLEs
τ
˜
1
,
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaacamaaBaaa
leaacaaIYaaabeaaaaa@3E96@
and
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acaaaa@3A39@
and the Bayesian
assisted estimators
τ
⌣
1
,
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaafamaaBaaa
leaacaaIYaaabeaaaaa@3EAE@
and
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afaaaa@3A45@
showed very
similar behavior which was characterized by serious problems of bias that
deteriorated their performance.
Notice that the percentages of samples in which the
proposed estimators were not computed because of numerical convergence
problems, as well as the number of samples in which the values of the
estimators exceeded
10
5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG
imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@
and that not
were used in calculating the reported results of
r
−
bias
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3B6C@
and
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca
qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@
because they
would have been seriously affected, were not large, except in Population IV. As
was indicated by a reviewer, computing the
r
−
bias
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3B6C@
and
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaab2gacaqGZbGaaeyzaaaa@3A8F@
of an estimator
using only its available values lower than
10
5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG
imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@
favors the
proposed estimators. We agree with that observation and for this reason we also
reported measures of the performance of the estimators based on the medians of
the relative errors and absolute relative errors which are robust to large
values of the estimators. Thus, if we supposed that any time that an estimator
was not computed its value had been very large and we computed the values of
the measures of the performance of the estimators that are based on the medians
using the complete set of observations the results would not have been
different from those reported in Table 6.2, and our conclusions based on these
measures would not have changed.
The results of the simulation study on the 95% CIs are
shown in Table 6.3. The main outcomes are the following. All the PLCIs and
adjusted PLCIs for
τ
1
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccaGG6aaaaa@3BD9@
the ones based
on the UMLE
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D85@
and those based
on the CMLE
τ
^
1
(
C
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaOGaaiilaaaa@3E2D@
showed good
values of the cp in Population I. The adjusted PLCIs presented also good values
of the cp in Population II, but not the unadjusted PLCIs whose values of the cp
were relatively low. In Population III the values of the cp of all the PLCIs
and adjusted PLCIs for
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
were low,
whereas in Population IV the values were only slightly low. A good
characteristic of these CIs was that they showed pretty acceptable values of
their mrl and mdrl in each of the situations that were considered. The PLCI for
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
based on
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D86@
and the one
based on
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaaaa@3D74@
presented
acceptable values of the cp in all the populations, except in Population IV,
where the values were something low. However, in all the cases the mrl and mdrl
of these CIs were so large that they were not useful for making reasonable
inferences. Both PLCI for
τ
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDca
GG6aaaaa@3AE8@
the one based on
the UMLE
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaaaa
@3CCA@
and that based
on the CMLE
τ
^
(
C
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGa
aiilaaaa@3D72@
performed
acceptably well in Populations I and II, although the means of their relative
lengths were large in Population II because this measure is not robust to great
values of the lengths of the intervals. In the other populations these CIs
showed problems of low coverage and/or large relative lengths; thus their
performance was not good. Both types of adjusted PLCIs performed well only in
Population I, in the other populations they presented large values of their
relative lengths. Neither of the two types of CIs: the ones based on the UMLEs
and those based on CMLEs performed uniformly better than the other, but those
based on the UMLEs performed in a greater number of cases slightly better than
those based on the CMLEs.
With respect to the bootstrap CIs, we have that each of
the two types of CIs for
τ
1
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccaGG6aaaaa@3BD9@
the one based on
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaaaa@3D85@
and that based
on
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk
aaaaaaaa@3D73@
performed well
in Populations I, II and IV, although in this last population the values of their
cp were slightly low. In Population III the values of their cp were very low
because of the biases of the point estimators of
τ
1
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccaGGUaaaaa@3BCD@
The two types of
bootstrap CIs for
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
performed badly
in all the populations because the values of their relative lengths were large.
Finally, the two types of CIs for
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
performed in
general well. Notice that the values of their mrl tended to be large because
this measure is not robust to great values of the lengths, whereas the values
of their mdrl were acceptable. Neither of the two types of bootstrap CIs
performed uniformly better than the other, but the CIs based on the UMLEs
performed in most cases better than those based on the CMLEs.
Table 6.3
Coverage probabilities and means and medians of relative lengths of the 95% confidence intervals for the
population sizes
Table summary
This table displays the results of Coverage probabilities and means and medians of relative lengths of the 95% confidence intervals for the
population sizes. The information is grouped by Population
Sampling rates
Conf. Interval (appearing as row headers), I, II, III, IV, XXXX
0.51, XXXX, XXXX
0.40 and XXXX
0.50, calculated using cp, mrl and mdrl units of measure (appearing as column headers).
Population
I
II
III
IV
Sampling rates
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.51
This is an empty cell
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.50
This is an empty cell
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.51
This is an empty cell
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
f
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIXaaabeaaaaa@39D4@
0.51
This is an empty cell
f
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa
aaleaacaaIYaaabeaaaaa@39D5@
0.40
Conf. Interval
cp
mrl
mdrl
cp
mrl
mdrl
cp
mrl
mdrl
cp
mrl
mdrl
PLCI-
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
0.95
0.22
0.22
0.85
0.22
0.22
0.61
0.24
0.24
0.891.6
0.23
0.24
Adj-PLCI-
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
0.94
0.24
0.24
0.98
0.42
0.41
0.69
0.27
0.26
0.901.6
0.25
0.26
PLCI-
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D23@
0.94
1.4
0.98
0.95
2.82
1.3
0.95
2.6
1.6
0.7719
7.16
1.4
PLCI-
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@
0.95
0.66
0.59
0.97
0.91
0.65
1.0
1.0
0.79
0.8621
2.11
0.65
Adj-PLCI-
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@
0.92
0.75
0.62
1.07.0
5.833
2.1
1.00.20
1.71
0.87
0.9022
2.74
0.78
Bootstr-CI-
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
0.94
0.23
0.23
0.94
0.33
0.31
0.59
0.25
0.25
0.890.40
0.24
0.25
Bootstr-CI-
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D23@
0.87
1.4
0.86
0.97
3.9
1.6
0.97
4.7
2.1
0.8313
4.63
0.90
Bootstr-CI-
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@
0.93
0.38
0.28
0.99
0.97
0.49
0.98
1.1
0.52
0.8913
1.23
0.31
PLCI-
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.96
0.24
0.23
0.91
0.25
0.24
0.76
0.31
0.29
0.902.1
0.25
0.25
Adj-PLCI-
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.95
0.26
0.25
0.99
0.44
0.42
0.81
0.33
0.32
0.902.1
0.27
0.27
PLCI-
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D11@
0.94
1.4
0.98
0.95
2.72
1.3
0.95
2.5
1.6
0.8525
7.54
1.7
PLCI-
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
0.95
0.64
0.54
0.941.6
1.2
0.62
0.862.0
3.11
0.74
0.9330
2.91
0.84
Adj-PLCI-
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
0.96
0.82
0.59
1.07.6
7.233
2.6
0.903.8
3.66
1.2
0.9432
2.95
0.90
Bootstr-CI-
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.96
0.29
0.29
0.98
0.31
0.31
0.48
0.40
0.39
0.891.6
0.31
0.30
Bootstr-CI-
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D11@
0.90
1.5
1.0
0.98
4.7
1.7
0.97
5.5
2.4
0.9424
3.46
1.3
Bootstr-CI-
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
0.95
0.44
0.34
1.00
1.1
0.49
0.96
1.3
0.62
0.9425
0.886
0.43
Wald-CI-
τ
˜
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@
0.00
0.09
0.09
0.00
0.08
0.08
0.00
0.08
0.08
0.08
0.13
0.13
Wald-CI-
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@
0.00
0.17
0.16
0.00
0.17
0.16
0.00
0.16
0.16
0.18
0.34
0.31
Wald-CI-
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aaaaa@39D6@
0.00
0.08
0.08
0.00
0.07
0.07
0.00
0.07
0.07
0.03
0.13
0.13
Wald-CI-
τ
⌣
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@
0.00
0.09
0.09
0.00
0.09
0.09
0.00
0.08
0.08
0.11
0.15
0.15
Wald-CI-
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@
0.00
0.17
0.17
0.00
0.17
0.17
0.00
0.17
0.17
0.25
0.36
0.33
Wald-CI-
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
baaaa@39E2@
0.00
0.08
0.08
0.00
0.08
0.08
0.00
0.08
0.07
0.04
0.15
0.14
With regard to the CIs based on the point estimators
derived under the homogeneity assumption, all of them showed null values of the
cp, except in Population IV where the values were different from zero, but
still too low. The bad performance of these CIs in terms of the cp was a result
of the large biases of the point estimators. Thus, despite of the very small
values of the
r
−
ml
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaab2gacaqGSbaaaa@39A0@
of these
intervals the very low values of their cp did not allow making reasonable
inferences.
Observe that in the first three populations the
percentages of samples in which the proposed CIs were not computed because of
numerical convergence problems as well as the number of samples in which the
values of their relative lengths exceeded
10
5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG
imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@
and that not
were used in calculating the reported results of the mrl because they would
have been seriously affected, were not large (less than 4%), except in the case
of the adjusted PLCIs based on
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaaaa
@3CCA@
and on
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaaaa
@3CB8@
which in
Population II were not computed in about 7% of the samples and the means of
their relative lengths were computed without using 33 values greater than
10
5
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG
imamaaCaaaleqabaGaaGynaaaakiaac6caaaa@3B82@
However, in
Population IV some percentages were close to 20% and others close to 30%. The
large values of these percentages were, in part, consequence of the relatively
large values of the percentages of samples in which the corresponding point
estimators were not computed. It is clear that computing the measures of
performance of a CI using only the cases in which the interval was obtained or
computing the rml using only the samples in which the relative lengths were
lower than
10
5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG
imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@
favors the
proposed CIs. However, notice that practically in all cases in which those
percentages were large, say larger than 5%, both the mrl and the mdrl of the
intervals were large enough that those intervals were not useful for making
inferences. Therefore, if the performance of these CIs was not good under this
favorable assessment, it will not be good under a fairer evaluation. The
exceptions to this pattern were the PLCI based on
τ
^
1
(
U
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk
aaaaaOGaaiilaaaa@3E3F@
and the two
types of bootstrap CIs for
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
which showed
acceptable performance and large percentage of samples in which were not
computed. So, the results of these CIs should be taken with reserve.
6.2 Population constructed using data from the
Colorado Springs study on HIV/AIDS transmission
In this simulation study we constructed a population
using data from the Colorado Springs study on heterosexual transmission of
HIV/AIDS. As was indicated in the introduction to this paper, this
epidemiological research was focused on a population of people who lived in the
Colorado Springs metropolitan area from
1982
−
1992
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeymaiaabM
dacaqG4aGaaeOmaiabgkHiTiaabgdacaqG5aGaaeyoaiaabkdaaaa@3C8D@
and who were at
high risk of acquiring and transmitting HIV. That population included drug
users, sex workers and their personal contacts, defined as those persons with whom
they had close social, sexual or drug-associated relations. In that study, 595
initial responders were selected in a non-random fashion through a sexually
transmitted disease clinic, a drug clinic, self-referral and street outreach.
The responders were asked for a complete enumeration of their personal contacts
and a total of 7,379 contacts who were not in the set of the initial responders
were named and included in the study. In our simulation study the set
U
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS
baaSqaaiaaigdaaeqaaaaa@3A26@
was defined as
the set of the 595 initial responders and, as in Félix-Medina and Monjardin
(2010), they were grouped into
N = 105
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0JaaGymaiaaicdacaaI1aaaaa@3C72@
clusters of
sizes
m
i
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS
baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C34@
generated by
sampling from a zero-truncated negative binomial distribution with parameter of
size 2.5 and probability
2
/
3
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai
aaikdaaeaacaaIZaaaaiaac6caaaa@3AA6@
The sample mean
and variance of the 105 values
m
i
’
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS
baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C34@
were 5.67 and 15.03,
respectively. It is worth noting that most of the people who were assigned to
the same cluster came from the same original source of recruitment. A person
was defined to be linked to a cluster if he or she was a personal contact of at
least one element in that cluster. Since, approximately 95% of the 7,379
contacts of the initial responders were linked to only one cluster, and this
could affect the performance of the proposed estimators, in our study we
defined the set
U
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS
baaSqaaiaaikdaaeqaaaaa@3A27@
as the subset of
the 7,379 contacts formed by the 415 persons who were linked to at least two
clusters plus the 379 sex workers who were linked to only one cluster. Thus,
τ
1
= 595 ,
τ
2
= 794
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaI1aGaaGyoaiaaiwdacaGG
SaGaeqiXdq3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaG4naiaaiM
dacaaI0aaaaa@4511@
and
τ = 1,389 .
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq
GH9aqpcaqGXaGaaeilaiaabodacaqG4aGaaeyoaiaac6caaaa@3F72@
It is worth
noting that this population is the same as the one called “reduced population”
by Félix-Medina and Monjardin (2010).
We set the sizes of the initial samples selected from
the population to
n = 25.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey
ypa0JaaGOmaiaaiwdacaGGUaaaaa@3C8B@
This value of
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3958@
yielded the
sampling rates:
f
1
= 0.46
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaiaac6cacaaI0aGaaGOn
aaaa@3E31@
and
f
2
= 0.37.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS
baaSqaaiaaikdaaeqaaOGaeyypa0JaaGimaiaac6cacaaIZaGaaG4n
aiaac6caaaa@3EE4@
The simulation
experiment was carried out as the previous one, except that each time that the
value
m
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS
baaSqaaiaadMgaaeqaaaaa@3A71@
was contained in
an initial sample, all the people linked to cluster
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@3953@
were included in
the sample. We used the same number of replications
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbaaaa@395C@
and the same
number
B
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGcbaaaa@392C@
of bootstrap
samples as those used in the previous study. In addition, the values of the
parameters of the initial distributions that were used to construct the
Bayesian-assisted estimators
τ
⌣
k
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afamaaBaaaleaacaWGRbaabeaaaaa@3B61@
and the value of
q
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@395B@
used to compute
the Gaussian quadrature formulas (3.3) and (3.4) were the same as those used in
the previous study.
The results of the simulation study are shown in Table
6.4. We can see that among the proposed estimators of the population sizes, only
the estimators of
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
did not present
problems of bias nor problems of instability. The estimators of
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
exhibited
serious problems of bias, particularly the estimators of
τ
2
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaGccaGGSaaaaa@3BCC@
which affected
their performance. As a result of the performance of the point estimators, only
the adjusted PLCIs and bootstrap CIs for
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
performed
acceptably well, although the values of the cp of the bootstrap CIs were
slightly low. The unadjusted PLCIs for
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
showed low
values of the cp because of the deviation from the assumed Poisson distribution
of the
M
i
’
s
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohacaGGUaaaaa@3CC6@
The PLCIs and
bootstrap CIs for
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
presented very
large values of the mrl and mdrl that these intervals were not useful. Observe
that the percentages of samples in which the proposed point and interval
estimators of
τ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGymaaqabaaaaa@3B11@
were not
computed because of numerical convergence problems were small (less than 1.2%).
Therefore, they were virtually not favored by the evaluation procedure. In the
case of the proposed point and intervals estimators of
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
those
percentages were large. However, if their performance was not good under this
favorable assessment, it will not be good under a fairer evaluation.
With regard to the point estimators derived under the
homogeneity assumption, we have that the MLEs
τ
˜
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIXaaabeaaaaa@3B20@
and
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIYaaabeaaaaa@3B21@
showed problems
of bias which affected their performance; however, the estimator
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acaaaa@3A39@
did not show
problems of bias and its performance was acceptable. The small bias exhibited
by this estimator might be explained by the fact that the negative bias of
τ
˜
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIXaaabeaaaaa@3B20@
was canceled out
by the positive bias of
τ
˜
2
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
acamaaBaaaleaacaaIYaaabeaakiaac6caaaa@3BDD@
The
Bayesian-assisted estimators performed similarly to the previous ones, although
in this case the estimator
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga
afamaaBaaaleaacaaIYaaabeaaaaa@3B2D@
of
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
showed only mild
problems of bias. The Wald CIs based on the MLEs and on the Bayesian-assisted
estimators showed low values of the cp. However, since the values of the
r
−
ml
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk
HiTiaab2gacaqGSbaaaa@39A0@
of these
intervals were acceptable, the intervals for
τ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda
WgaaWcbaGaaGOmaaqabaaaaa@3B12@
and
τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa
a@3A2A@
might provide
some information about these parameters.
Table 6.4
Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Point estimators
Table summary
This table displays the results of Résultats Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Point estimators. The information is grouped by Estimators (appearing as row headers) and Point estimators(appearing as column headers).
Estimator
r
−
bias
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbeqabeWaceGabiqabeqabmqabeabbaGcbaGaaeOCaiabgk
HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3D98@
r
−
mse
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x
fr=xfbpdbeqabeWaceGabiqabeqabmqabeabbaGcbaWaaOaaaeaaca
qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3CD6@
mdre
mdare
Uncond. heter. MLEs
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
-0.00
1
0.03
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc
dacaqGSaGaaeimaiaabcdadaqhaaWcbaGaaGymaaqaaiaabcdacaqG
SaGaaeimaiaabodaaaaaaa@3F2D@
0.10
-0.01
0.07
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D23@
1.7
16
3.5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeymaiaabY
cacaqG3aWaa0baaSqaaiaaigdacaaI2aaabaGaae4maiaabYcacaqG
1aaaaaaa@3DE4@
4.5
0.79
0.79
τ
^
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaaiaaiIcacaWGvbGaaGykaaaaaaa@3C43@
0.95
17
3.5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY
cacaqG5aGaaeynamaaDaaaleaacaaIXaGaaG4naaqaaiaabodacaqG
SaGaaeynaaaaaaa@3E9E@
2.6
0.46
0.46
Cond. heter. MLEs
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.01
0.57
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY
cacaqGWaGaaeymamaaDaaaleaaaeaacaqGWaGaaeilaiaabwdacaqG
3aaaaaaa@3DCB@
0.12
0.01
0.08
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D11@
1.7
10
4.7
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeymaiaabY
cacaqG3aWaa0baaSqaaiaaigdacaaIWaaabaGaaeinaiaabYcacaqG
3aaaaaaa@3DE0@
4.5
0.80
0.80
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
0.96
10
5.2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY
cacaqG5aGaaeOnamaaDaaaleaacaaIXaGaaGimaaqaaiaabwdacaqG
SaGaaeOmaaaaaaa@3E96@
2.6
0.46
0.46
Homogeneous MLEs
τ
˜
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@
-0.22
0.23
-0.22
0.22
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@
0.21
0.34
0.16
0.18
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aaaaa@39D6@
0.02
0.17
-0.00
0.10
Homogeneous BEs
τ
⌣
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@
-0.22
0.23
-0.22
0.22
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@
0.12
0.22
0.10
0.13
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
baaaa@39E2@
-0.02
0.13
-0.04
0.09
Table 6.4(cont.)
Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Confidence intervals
Table summary
This table displays the results of Résultats Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Confidence intervals. The information is grouped by Conf. interval (appearing as row headers) and Confidence intervals(appearing as column headers).
Conf. interval
cp
mdre
mdare
PLCI-
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
0.75
0.24
0.24
Adj-PLCI-
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
0.95
0.41
0.41
PLCI-
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D23@
0.398.3
1018
3.7
PLCI-
τ
^
( U )
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaaiaacIcacaWGvbGaaiykaaaaaaa@3C37@
0.838.6
5.76
2.1
Adj-PLCI-
τ
^
( U )
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaaiaacIcacaWGvbGaaiykaaaaaaa@3C37@
0.9921
1147
7.5
Bootstr-CI-
τ
^
1
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D22@
0.91
0.37
0.37
Bootstr-CI-
τ
^
2
(
U
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca
aaaaaaa@3D23@
0.863.2
1129
3.6
Bootstr-CI-
τ
^
( U )
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaaiaacIcacaWGvbGaaiykaaaaaaa@3C37@
0.883.2
6.328
2.0
PLCI-
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.811.2
0.30
0.27
Adj-PLCI-
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.971.2
0.45
0.44
PLCI-
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D11@
0.3910
9.617
3.6
PLCI-
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
0.8920
6.29
2.6
Adj-PLCI-
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
1.027
1432
9.3
Bootstr-CI-
τ
^
1
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D10@
0.861.2
0.35
0.35
Bootstr-CI-
τ
^
2
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca
aaaaaaa@3D11@
0.908.0
9.735
3.9
Bootstr-CI-
τ
^
(
C
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK
aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@
0.919.2
5.934
2.2
Wald-CI-
τ
˜
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@
0.06
0.16
0.16
Wald-CI-
τ
˜
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@
0.71
0.60
0.53
Wald-CI-
τ
˜
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG
aaaaa@39D6@
0.73
0.35
0.31
Wald-CI-
τ
⌣
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@
0.13
0.22
0.22
Wald-CI-
τ
⌣
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@
0.72
0.45
0.43
Wald-CI-
τ
⌣
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9
vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq
baaaa@39E2@
0.70
0.27
0.26
Editorial policy
Survey Methodology publishes articles dealing with various aspects of statistical development relevant to a statistical agency, such as design issues in the context of practical constraints, use of different data sources and collection techniques, total survey error, survey evaluation, research in survey methodology, time series analysis, seasonal adjustment, demographic studies, data integration, estimation and data analysis methods, and general survey systems development. The emphasis is placed on the development and evaluation of specific methodologies as applied to data collection or the data themselves. All papers will be refereed. However, the authors retain full responsibility for the contents of their papers and opinions expressed are not necessarily those of the Editorial Board or of Statistics Canada.
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Copyright
Published by authority of the Minister responsible for Statistics Canada.
© Minister of Industry, 2015
Catalogue no. 12-001-X
Frequency: semi-annual
Ottawa
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Date modified:
2017-09-20